Aplications - 9.24gb
Library - 31.12gb
there should be more then 300gig free
Those are visible folders - there's a whole Unix underpinning which won't be displayed in the finder, including swapfiles, caches, logfiles and all kinds of other stuff. MacOS does manage this stuff (eg archive log files etc), but it does also take up space.
If you are not sure where your files are that are taking up space, you might want to try one of the "Cleaner" type programs, like CleanMyMac, which will help you optimise this stuff.